Photonic Bandgap Crystals (hereinafter PBC) are periodically structured materials characterized by ranges of frequency in which light cannot propagate through the structure. A PBC is generally composed of at least two materials differing in their refraction indexes, whose periodicity is approximately half the wavelength of light unable to propagate through the lattice. A complete photonic band gap is a range of frequencies in which there are no real solutions of Maxwell's equations. The theoretical treatment of the interaction of such periodic materials with electromagnetic radiation is analogous to the treatment in solid-state physics of the interaction of atomic lattices with electrons, showing the existence of distinct energy bands separated by gaps.
As actual photonic crystals are complex structures, in general it is hard to locate the bandgaps analytically. One method entails finite difference numerical analysis, calculating the evolution in time of the electromagnetic fields, thus extracting measurable quantities such as transmission or reflection. Another method consists in diagonalizing the Magnetic field Hermitian operator in the frequency domain, thus finding its eigenvectors, the allowed modes and extracting the band diagrams. See “Introduction to Photonic Crystals: Bloch's Theorem, Band Diagrams, and Gaps” by Steven G. Johnson and J. D. Joannopoulos, MIT 2003. 2D Photonic Crystal slabs and their bandgaps as a function of the dimensions and refractive indexes of the composing materials have been extensively investigated, see for example “Intrinsic eigenstate spectrum of planar multilayer stacks of two-dimensional photonic crystals” by K. H. Dridi Optics Express 2003/Vol. 11, No. 10/p 1156.
Simple two-dimensional PBCs structured for example by periodic holes in a high refractive index matrix, can exhibit polarization-dependent bandgaps, in the frequency (w=a/λ) versus the wavenumber (k) domain. These crystals, though, cannot confine electromagnetic radiation in the third dimension, due to the “leaking” of the transverse electromagnetic waves into the 3rd dimension. Limiting the “thickness” of the slab and constricting the slab between two air boundaries, or low refraction index materials, can minimize the “leakage” into the third dimension. There are two basic topologies for 2D photonic crystals, low-index holes in a high index matrix or high index rods in a low index matrix.
The largest bandgaps in 2D PBG crystals are obtained with triangular lattices with very large, almost touching holes and when the dielectric constant contrast is large, ε˜12 for example. Photonic Bandgap structures having high contrast refractive indexes, effectively act as mirrors. Moreover PBCs modify the spontaneous emission rate of excited atoms within the lattice, when the imbedded atom has an emission frequency in the bandgap. The absence of electromagnetic modes inside the photonic bandgap, causes atoms or molecules imbedded in the crystal, to be locked in an excited state relative to the ground state. Photonic Bandgaps can be switched on and off by rapidly changing the refractive index of a non-linear component of the PBC, electrically or optically.
A software program for computing the band structures (dispersion relations) and electromagnetic modes of periodic dielectric structures (the MIT Photonic-Bands MPB) is freely available for download from the www.ab-initio.mit.edu/mpb website. Photonic Crystal design tools are also available commercially, for example from Photon Design ltd. Oxford, UK.
Gamma ray radiation such as, the 140 keV γ-rays emitted from technetium-99m or the 511 keV γ-rays emitted following the annihilation of positrons emitted from fluorine-18, are usually detected by scintillation crystal detectors such as NaI (Tl), CsI (Tl), Bi4GeO4 or CdWO4 scintillators or semiconductor detectors such as Ge or CdZT. The γ rays interact with the atoms of the crystal through the Photo-electric or Compton effects and Pair generation processes. Table 1 gives the energy and atomic number dependency of the 3 processes.
TABLE 1Type ofZ dependenceZ dependenceInteractionEnergy dependence(cm2/atom)(cm2/g)PhotoelectricE−3.5Z4 to 5Z3 to 4ComptonE−0.5 to −1Z~Z independentPair productionE to ln EZ2Z
The energy transfer from a hard photon (hν) to an atomic electron (e) in the photoelectric effect is given by Ee=hν−Eb where Eb is the binding energy of the electrons of the stopping material. The atom excited by the stripping of one of its electrons, returns to its stable state by emitting one or more X-rays whose energies are determined by its discrete energy levels and denoted accordingly as the M, L or K X-rays.
In the Compton effect which is effective at higher energies and low Z elements, the incoming photons are scattered by the free electrons of the stopping material, imparting them part of their energy. The energy of the scattered gamma ray (1) is given byE1=hν′ mec2/[1+cosθ+(mec2/E0)]or cosθ=1−(mec2/Ee)+(mec2/E1+Ee)the recoil electron's energy is given byEe=hν′=[(hν)2/mec2(1−cosθ)]/[1+(hν/mec2)(1−cosθ)]
The maximal energy of the recoil electron is therefore at Eemax=E/(1+mec2/2E)
The recoil angle φ of the electron, relative to the direction of the impinging gamma ray, is given bycotφ=1+(hν/mec2)tan(θ/2)
It is important to note that following the momentum equalities, the incoming gamma ray, the scattered gamma ray and the recoil electron are all on the same plane.
The differential cross section of the Compton Scattering for unpolarized photons is given by the Klein-Nishina equation:[dσ/dΩ]=(re2/2)(ν′/ν)2[(ν/ν′)+(ν′/ν)−sin2θ]where re=(e2/mec2) is the “classical” radius of the electron equal to 2.82 10−13 cm
This equation which assumes scattering by free electrons, has to be modified by a form factor S(k,k′) at energies where the binding energies of the electrons become important, as compared with the energy of the gamma ray, causing the angular distribution in the forward direction to be suppressed.
In the pair production effect the two 511 Kev hard photons generated by the annihilation of the positron, again interact with the stopping material through the Compton or Photoelectric effects and cause ejection of electrons and their eventual absorption, as explained above.
The electrons so produced by the three processes are stopped in the crystal producing low energy photons in the visible range, along their track. If the crystal is transparent to these photons, they can emerge from the crystal and be detected by a photon detector such as a photomultiplier tube or a photo-diode.
The range of the knoked-off electron in the crystal can be obtained by observing that
      R    ⁢                  ⁢          (              E        0            )        =                    ∫                  E          0                0            ⁢                        (                                    ⅆ              x                        /                          ⅆ              E                                )                ⁢                                  ⁢                  ⅆ          E                      =                  -                              ∫            0                          E              0                                ⁢                                                    (                                                      ∂                    E                                    /                                      ∂                    x                                                  )                                            -                1                                      ⁢                                                  ⁢                          ⅆ              E                                          =                        ∫          0                      E            0                          ⁢                              (                          1              /              S                        )                    ⁢                                          ⁢                      ⅆ            E                              and using the Bethe-Bloch equation for electron absorption in matterdE/dx=C[1/v]NZ ln{f(I,v)},where C is an empirical constantv=velocity of incident particle; N=number of atoms per cm3; Z=atomic number of absorbing medium; I=the mean excitation energy; f=a function of I and v.
The electron's range in low Z materials and in the 0.01≦T≦2.5 MeV range may be approximated by the empirical formulaR=0.412T1.27−0.0954lnT where R is the range in g/cm2 units and T the kinetic energy of the electron in units of MeV
The length of part of an electron track, may be calculated by observing that E=∫(∂E/∂x)dx; therefore the energy absorbed in part of the track is
      E    0    =                              (                                    ∂              E                        /                          ∂              x                                )                avg            ⁢                        ∫                      R            0                          ⁢                                  ⁢                  ⅆ                                          ⁢          x                      =                                        (                                          ∂                E                            /                              ∂                x                                      )                    avg                ⁢                                  ⁢                  R          0                ⁢                                  ⁢        giving        ⁢                                  ⁢        for        ⁢                                  ⁢                  R          0                    =                        E          0                /                                                            (                                                      ∂                    E                                    /                                      ∂                    x                                                  )                            avg                                      (                              R                0                            )                                .                    
Thus if the energy absorbed in this part of the track E0 and the average differential absorption (∂E/∂x) are known, the length of the portion of the track (R0) may be calculated.
The underlying principles of Compton Cameras, namely that knowing the energy of the scattered gamma ray E1, the position of the subsequent (usually photo-electric) event and the energy of the recoil electron and the position of its track, enable to derive the direction of the incoming gamma ray, are self evident and follow from the Compton effect equations. The practical obstacles are the measurement of the electron's position before the scattering and the direction of its recoil after interaction with the incoming gamma ray. Numerous variations of two-detector geometries and schemes have been proposed, that consist in detecting the position of the gamma-electron interaction position with one detector and that of the subsequent photoelectric event with the second detector, in addition to their energies, in order to determine the direction of the incoming gamma ray on the “scattering plane”. As the absolute direction of the “scattering plane” in space remains unknown, the absolute direction of the incoming gamma ray remains ambiguous and within a “cone” whose aperture is the scattered gamma ray's scattering angle relative to the incoming gamma ray in space.
In medical physics applications, as the “gamma ray-to-detector” distance is small (an approximation that is certainly wrong in case of astrophysics geometries), knowledge of the approximate direction of the incoming gamma ray, up to a cone, enables to approximately locate the position of the gamma source; moreover looking at the intersection of a number of “cones” enables to reduce the indeterminacy of the source position. A further reduction of this indeterminacy has been proposed by combining multiple pinhole collimators and the “Compton direction up to a cone” that consists in eliminating for each event, the pinholes that do not fall within the “cone”, thus benefiting from the increased efficiency of multiple pinholes.
An additional uncertainty as to the direction of the recoil electron arises from the fact that the “free” electron is not at rest. If the electron's “motion” is taken into account, its recoil direction will not be exactly in the “scattering plane” but slightly out of it. This effect is sometimes referred to as the “doppler brodening” of the recoil electron's direction.
Radiation detectors are used in several healthcare instruments to map the radiation emitted by the body of a patient which has previously been injected with a radionuclide. Gamma Cameras endeavor to detect the location of the radiating source, by collimating the incoming radiation and then detecting the points of interaction of the collimated gamma rays with the radiation detector. Straight tubular collimators give a map of the radionuclides in the body of the patient, while pinhole collimators give the emission intensity coming from the focal point of the collimator, at the expense of blocking anything else. High resolution collimators immensely reduce the radiation intensity detected by the gamma camera and greatly increase the time needed for mapping the radionuclide distribution with acceptable spatial resolutions.
Gamma cameras with straight tubular collimators give a 2D projection of the radionuclide distribution, on the surface of the camera and cannot give depth information. SPECT (Single Photon Emission Computer Tomography) consists in rotating the Gamma Camera, to obtain several 2D maps from multiple directions and reconstruct the distribution along the third dimension, thus enabling to obtain a Tomographical image of the radionuclide distribution. Obviously dispensing with the collimator in SPECT is also of great benefit, reducing the scanning time or the amount of radionuclide injected or both.
PET (Photon Emission Tomography) consists in injecting the patient with a positron emitting radionuclide and detecting the annihilation pair of 511 keV gamma rays emitted in opposite directions, using two radiation detectors in time coincidence, thus determining a line that crosses the position of the radionuclide source. Detecting a multiplicity of such pairs of 511 keV gamma rays gives lines that crisscross the radiation source and enables to determine its position. The accuracy of the line crossing the radionuclide is determined both by the spatial extensions of the detectors and the exact location within the detector of the interacting gamma ray. Thus there is a need for a high efficiency and high spatial resolution gamma detector.
Cross section images of a body (slices) can be imaged by Computerized Tomography (CT) consisting in irradiating the body with intense X-rays from all 180° or more angles, detecting the unattenuated X rays that traversed the body using radiation detectors, and reconstructing a density map of the slice of the body traversed by the X rays, that is consistent with the integral absorption data obtained from all angles. The spatial resolution of such density images and the thickness of the slices, are dependent, inter alia, also on the spatial resolution of the radiation detectors. Multiple adjacent slices require thin adjacent detectors with minimal separating walls. A high spatial resolution continuous radiation detector therefore enables imaging a continuous multiplicity of thin slices and thus obtaining a continuous volumetric image.
Search of the previous art has not revealed any patent, patent application or publication that deals with scintillator fiber arrays structured as Photonic Bandgap Crystals, nor scintillator fiber arrays with an ability to find the position of the scintillation along the fiber.